328 Prof. R. Clausius on the Application of a 



Secondly, it shall be assumed that n = — 2, which corresponds 

 to Newton's law of attraction, that governs the motion of the 

 heavenly bodies. For this case the preceding equations become : — * 



-k-=-k- = 2B, (35) 



r p v 



_^ = -.- = _E, (36; 



2 = 27r\/y.pf=27rAVm(-2E)" 1 . . (37) 

 The last equation, which may also be written 



corresponds to Kepler's third law, which is contained in our 

 equations as a special case. It must, however, be somewhat 

 otherwise expressed than it was by Kepler and frequently still 

 is, namely that the squares of the periods are as the cubes of the 

 mean distances. This is not strictly correct; for p is not the 



mean value of r. but - is the mean value of -. The other, more 

 p r 



accurate form of the theorem, that the squares of the periods are 



as the cubes of the major axes of the ellipses, agrees perfectly with 



our equation ; for it can easily be proved that p is equal to half 



the major axis of the ellipse which forms the path when the 



central force is of this kind. 



We now turn to the motion of two material points about each 

 other. 



Let us first assume that we have any number of material 

 points which move in a stationary manner in closed paths, and 

 that these motions undergo an infinitely small alteration, so that 

 stationary motions in closed paths again arise ; my equation reads 

 for this case: — 



m rT7> 



-2(X^ + YSz/ + Z^) = 2:^3?; 2 + 2m ? ; 2 Slogi. . (38) 



If the forces operating on the points are such that they have an 

 ergal, which we will denote by U, and if we again suppose that 

 with the alteration of the motion the ergal remains an invariable 

 function of the coordinates of all the points, then we can put, 

 corresponding to equation (3) : — 



BJJ=X~^ +W>Slog2. . . . (39) 



If the forces operating in our system consist only of the attrac- 



